#446241

order	512
nilpotency class	3
isoclinism	isoclinic class of #446240
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15836
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

846 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2)(3,14,18,25)(4,19)(5,11)(6,12)(7,13)(8,16,10,17)(9,21)(15,24,30,27)(20,26,31,32)(22,29)(23,28);(1,3,5,16,7,18,6,17)(2,8,12,25,13,10,11,14)(4,20,21,30,23,31,22,15)(9,26,19,24,29,32,28,27);(1,4,5,21,7,23,6,22)(2,9,12,19,13,29,11,28)(3,15,17,31,18,30,16,20)(8,24,14,26,10,27,25,32);(1,3)(2,10,13,8)(4,15)(5,17)(6,16)(7,18)(9,27,29,24)(11,14,12,25)(19,32,28,26)(20,22)(21,31)(23,30)

Polycyclic presentation:

2^9|a^2=eg,b^2=f,c^2=efg,d^2=efi,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,g^b=gi,h^b=hi,h^c=hi,e^d=ei,f^d=fi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 353437);